2015/11/17 by Xi-rong Dai, Hui Rao, Dai, Xi-rong +3
Mathematics · #27A05 #28A80 #Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN) #math.DS #math.GN #msc:27A05 #msc:28A80
paper · pdf · doi:10.48550/arxiv.1511.05411
44 pages, 16 figures
arxiv created 2018/04/23 · arxiv updated 2018/04/24
It is well-known that the constructions of space-filling curves depend on certain substitution rules. For a given self-similar set, finding such rules is somehow mysterious, and it is the main concern of the present paper. Our first idea is to introduce the notion of skeleton for a self-similar set. Then, from a skeleton, we construct several graphs, define edge-to-trail substitution rules, and explore conditions ensuring the rules lead to space-filling curves. Thirdly, we summarize the classical constructions of the space-filling curves into two classes: the traveling-trail class and the positive Euler-tour class. Finally, we propose a general Euler-tour method, using which we show that if a self-similar set satisfies the open set condition and possesses a skeleton, then space-filling curves can be constructed. Especially, all connected self-similar sets of finite type fall into this class. Our study actually provides an algorithm to construct space-filling curves of self-similar sets.